## Abstract

We investigate some kinetic properties of an isotopically disordered harmonic crystal. We prove rigorously that for almost all disordered chains the transmission coefficient of a plane wave with frequency ω, t N(ω), decays exponentially in N, the length of the disordered chain, with the decay constant proportional to ω^{2} for small ω. The response of this system to an incident wave is related to the nature of the heat flux J(N) in a disordered chain of length N placed between heat reservoirs whose temperatures differ by ΔT>0. We clarify the relationship between the works of various authors in the heat conduction problem and establish that for all models J(N)→0 as N→∞ in a disordered system. The exact asymptotic dependence of J(N) on N eludes us, however. We also investigate the heat flow in a simple stochastic model for which Fourier's law is shown to hold. Similar results are proven for two-dimensional systems disordered in one direction.

Original language | English (US) |
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Pages (from-to) | 692-703 |

Number of pages | 12 |

Journal | Journal of Mathematical Physics |

Volume | 15 |

Issue number | 6 |

DOIs | |

State | Published - 1973 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics